Abel-Jacobi map, integral Hodge classes and decomposition of the diagonal

TL;DR

This paper investigates the Abel-Jacobi map for smooth projective 3-folds, exploring the existence of special 1-cycle families, their relation to homological decompositions, and implications for the Hodge conjecture in specific fibrations.

Contribution

It establishes conditions for the surjectivity of the Abel-Jacobi map with rationally connected fibers and links these to integral homological decompositions of the diagonal, advancing understanding of Hodge classes.

Findings

Existence of 1-cycle families with surjective Abel-Jacobi maps and rationally connected fibers.

Connection between rationally connected 3-folds and homological decomposition of the diagonal.

Verification of the Hodge conjecture for certain fibrations into cubic threefolds.

Abstract

Given a smooth projective 3-fold Y, with $H^{3,0}(Y)=0$, the Abel-Jacobi map induces a morphism from each smooth variety parameterizing 1-cycles in Y to the intermediate Jacobian J(Y). We study in this paper the existence of families of 1-cycles in Y for which this induced morphism is surjective with rationally connected general fiber, and various applications of this property. When Y itself is rationally connected, we relate this property to the existence of an integral homological decomposition of the diagonal. We also study this property for cubic threefolds, completing the work of Iliev-Markoushevich. We then conclude that the Hodge conjecture holds for degree 4 integral Hodge classes on fibrations into cubic threefolds over curves, with restriction on singular fibers.